If I roll a dice (di?) with the numbers 1-6 on it an infinite number of times, am I guarenteed to roll a each number?
I think that if I roll it for an actual number (as in 100, 100000, 1000000000000 etc.) of times instead of infinitely, then the odds of not rolling each number are incredibly low, but not infinitely small. So, if I roll it an infinite number of times, does this make the odds of my not rolling a certain number infinetely small?
If so, do infinitely small odds mean that an event is guarenteed to happen?
Last, if I am guarenteed to roll each number, then doesn’t this also mean that I will roll each number an infinite number of times?
James Michael Foster, I believe the answer to your question, my friend, is simply that no, you can never GUARANTEE that each number will be rolled, however, the odds are so unbelievably low that you WON’T roll a number when throwing the die an infinite number of times that it mathematically is equivalent to the odds of having Andrew Mauldin answer your thread…
If you roll the die an infinite number of times, the probably that you will see each number at least once is exactly 1. However, it can still happen that you don’t see a particular number (but the probability of this event is 0). So in answer to your question: no, it’s not guaranteed, but don’t place any bets on it.
It’s important to keep in mind, when discussing thought experiments like this, that there are outcomes to this experiment which don’t involve you ever rolling, say, a six. You could even roll nothing but ones, forever; that outcome is just as likely as any other individual sequence of results. This is true even though the probability of rolling nothing but ones, forever, is technically zero. It could still happen.
Wait, it seems to me that this question is similar to Xeno’s paradox. And it seems to me that (5/6)^(infinity)=0. That means that the chance of never rolling a, say, “2” is zero. With calculus we can handle these little infinities can’t we?
It would be more appropriate to say that the limit of (5/6)[sup]n[/sup] as n goes to infinity is zero. But yes, limits are the theoretical underpinnings of calculus.
Apparently, anything is possible. Even seemingly impossible things like newbies outposting those who have hung around for more than 7 times as long (even if they are striving to be a mod), or either of us mastering vB code.
Thank you for the responses, and I look foreward to consulting you in the future.
All right, I’m not an expert on things mathematical, but I am fairly deep in the subject. And I don’t buy that statement. I say the odds are zero that you would never roll a 1 (for example), i.e. it is impossible. In the same sense that .999rep is exactly equal to 1, .000rep is exactly equal to 0. Limits 101 in action.
All right, I have nothing to back this up but my admittedly amateur opinion. But I still claim it.
As the number of rolls goes to infinity, the chance of never rolling a one goes to zero.
As a practical matter, though, you can’t roll an infinite number of times…and I don’t mean because you’ll die, or the sun will go out. By the very nature of infinity, you’ll never get to a point where you can say, “Yup, I’m done! I’ve rolled the dice exactly an infinite number of times!” Therefore, no matter how many times you roll the dice, there’s always some small but positive chance that you won’t roll a 1. It’s getting smaller all the time, but it’s still positive.
Tom Stoppard made a comedy routine out of this in “Rosencranz and Guildenstern Are Dead.”
(So how is this different than Zeno’s paradox? One difference is that rolling a dice takes a finite amount of time each time you do it. On the other hand, the events in Zeno’s paradox take smaller and smaller amounts of time–as the distances the runner must cover get smaller and smaller, the time it takes to cover those distances also gets smaller and smaller as well. That’s the “trick” that enables the runner to actually get somewhere.)
As others have mentioned, a probability of zero is not synonymous with impossible, just as a probability of one is not synonymous with certain. Things like that become misleading when you’re dealing with an infinite sample space. It may help to see it in a different example:
Say you pick a real number at random from the interval [0,1] (we’ll use the Lebesgue probability measure; if you don’t know what that means, just think of it intuitively as “every number has just as good a chance as any other number to be picked”). OK, so for some examples–there’s a 1/2 chance the number picked is less than 0.5, there’s a 1/10 chance the number will be between 0.6 and 0.7, and so forth. What’s the probability that exactly 0.3 will be picked? Or 0.546543? What about any particular number? It should be clear that all of these probabilities are zero. So, when the number is picked, whatever number it is, that number had a zero chance of being picked, but obviously it wasn’t impossible for that number to be picked, since it just was picked.
Back to the dice analogy, same principle; any infinite string of dice rolls has probability zero of happening, obviously, however, when all is said and done, one of those zero probability infinite strings will have happened.
Im with lemur 100%. It’s really asking whether the universe is discrete or continuous. There isnt a correct theoretical answer, but the practical answer that Newton would have given you is 1. Cuz thats the limit of the equation.
No one is denying that the probability of never rolling a 1 is zero. We’re all agreeing on that. (At least, in this thread we are.)
But in an experiment with an infinite number of possible outcomes, it’s frequently the case that every outcome has a probability of zero. (And this is in fact one of those cases.) That obviously doesn’t mean that every outcome is impossible; something has to happen.
And the outcome “2, 2, 2, 2, 2, 2, 2, 2, …” is just as likely, in every possible sense of the word, as the outcome “3, 2, 5, 3, 4, 4, 1, 6, 1, …”. The second outcome just looks more “random” to our intuition.
Well, it’s a bit difficult to explain exactly what “probability 1” means without going into measure theory, and that’s not an area that I’m particularly comfortable with. It’s also not easy to understand, so you’ll have to pardon me if I skirt the issue.
So the difference between an event with probability 1 and an event that is guaranteed to happen is that the event with probability 1 might not happen.
When you’re dealing with a finite sample space (i.e., there are only a finite number of different things that could happen), an event with probability 1 is guaranteed to happen. It’s only when you introduce an infinite sample space that things go screwy. See Cabbage’s post for a really good example.
OK, I think I get the explanation. ALL possible dice roll sequences have probability zero, and yet one “must” occur.
Hmmmm. Well, perhaps this is proof that the ancient mathematicians were right and that dealing with zeros and infinities is likely to bring down the wrath of the gods on us mortals for tampering with what man was not meant to know.
